Solve [12/(sqrt{5}+sqrt{3})]+[18/(sqrt{5}-sqrt{3})]?

$$[ 12 / ( \sqrt { 5 } + \sqrt { 3 } ) ] + [ 18 / ( \sqrt { 5 } - \sqrt { 3 } ) ] ?$$

Evaluate

$$3\left(\sqrt{3}+5\sqrt{5}\right)\approx 38.737172085$$

Solution Steps

Rationalize the denominator of \frac{12}{\sqrt{5}+\sqrt{3}} by multiplying numerator and denominator by \sqrt{5}-\sqrt{3}.

$$\frac{12\left(\sqrt{5}-\sqrt{3}\right)}{\left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right)}+\frac{18}{\sqrt{5}-\sqrt{3}}$$

Consider \left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{12\left(\sqrt{5}-\sqrt{3}\right)}{\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}}+\frac{18}{\sqrt{5}-\sqrt{3}}$$

Square \sqrt{5}. Square \sqrt{3}.

$$\frac{12\left(\sqrt{5}-\sqrt{3}\right)}{5-3}+\frac{18}{\sqrt{5}-\sqrt{3}}$$

Subtract 3 from 5 to get 2.

$$\frac{12\left(\sqrt{5}-\sqrt{3}\right)}{2}+\frac{18}{\sqrt{5}-\sqrt{3}}$$

Divide 12\left(\sqrt{5}-\sqrt{3}\right) by 2 to get 6\left(\sqrt{5}-\sqrt{3}\right).

$$6\left(\sqrt{5}-\sqrt{3}\right)+\frac{18}{\sqrt{5}-\sqrt{3}}$$

Rationalize the denominator of \frac{18}{\sqrt{5}-\sqrt{3}} by multiplying numerator and denominator by \sqrt{5}+\sqrt{3}.

$$6\left(\sqrt{5}-\sqrt{3}\right)+\frac{18\left(\sqrt{5}+\sqrt{3}\right)}{\left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right)}$$

Consider \left(\sqrt{5}-\sqrt{3}\right)\left(\sqrt{5}+\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$6\left(\sqrt{5}-\sqrt{3}\right)+\frac{18\left(\sqrt{5}+\sqrt{3}\right)}{\left(\sqrt{5}\right)^{2}-\left(\sqrt{3}\right)^{2}}$$

Square \sqrt{5}. Square \sqrt{3}.

$$6\left(\sqrt{5}-\sqrt{3}\right)+\frac{18\left(\sqrt{5}+\sqrt{3}\right)}{5-3}$$

Subtract 3 from 5 to get 2.

$$6\left(\sqrt{5}-\sqrt{3}\right)+\frac{18\left(\sqrt{5}+\sqrt{3}\right)}{2}$$

Divide 18\left(\sqrt{5}+\sqrt{3}\right) by 2 to get 9\left(\sqrt{5}+\sqrt{3}\right).

$$6\left(\sqrt{5}-\sqrt{3}\right)+9\left(\sqrt{5}+\sqrt{3}\right)$$

Use the distributive property to multiply 6 by \sqrt{5}-\sqrt{3}.

$$6\sqrt{5}-6\sqrt{3}+9\left(\sqrt{5}+\sqrt{3}\right)$$

Use the distributive property to multiply 9 by \sqrt{5}+\sqrt{3}.

$$6\sqrt{5}-6\sqrt{3}+9\sqrt{5}+9\sqrt{3}$$

Combine 6\sqrt{5} and 9\sqrt{5} to get 15\sqrt{5}.

$$15\sqrt{5}-6\sqrt{3}+9\sqrt{3}$$

Combine -6\sqrt{3} and 9\sqrt{3} to get 3\sqrt{3}.

$$15\sqrt{5}+3\sqrt{3}$$

Factor

$$3 {(\sqrt{3} + 5 \sqrt{5})} = 38.737172085$$